
\begin{section}{Simple and Elliptic Surface Singularities}

\begin{defn}
A {\bf surface singularity} is a germ 

$$(S,0) = (\{ x \in \CC^{N} \colon f_{1}(x) = \dots = f_{k}(x) = 0 \}, 0)$$

where each $f_{i} \colon \CC^{N} \to \CC$ is a germ of a holomorphic function.
\end{defn}

\begin{defn}
A singularity of $(S,0)$ is a {\bf simple singularity} if and only if it is defined by one of the following equations:

\begin{equation}
A_{n} \colon x^{n+1} + y^2 + z^2 = 0.
\end{equation}

\begin{equation}
D_{n} \colon x^{n-1} + xy^2 + z^2 = 0.
\end{equation}

\begin{equation}
E_{6} \colon x^4 + y^3 + z^2 = 0.
\end{equation}

\begin{equation}
E_{7} \colon x^3y + y^3 + z^2 = 0.
\end{equation}

\begin{equation}
E_{8} \colon x^5 + y^3 + z^2 = 0.
\end{equation}
\end{defn}

\begin{defn}
A singularity of $(S,0)$ is a {\bf simple elliptic singularity} if and only if it is defined by one of the following equations:

\begin{equation}
\boxed{\tilde{E}_{6} \colon x^6 + y^3 + z^2 + \lambda xyz = 0.}
\end{equation}

\begin{equation}
\boxed{\tilde{E}_{7} \colon x^4 + y^4 + z^2 + \lambda xyz = 0.}
\end{equation}

\begin{equation}
\boxed{\tilde{E}_{8} \colon x^3 + y^2 + \lambda xyz = 0.}
\end{equation}

Where $\lambda \in \CC$ such that these equations define an isolated singularity.
\end{defn}

\begin{rmk}
These surface singularities are obtained via Lie Algebras, and the dual graph of the minimal resolution $\pi \colon X \to (S,0)$ (where $X$ is a nonsingular surface, the map $\pi$ is proper and an isomorphism when restricted to $\pi^{-1}(S - 0)$) is the Dynkin diagram of the same name.
\end{rmk}

\end{section}
